The function f is defined by \(f(x) = 1 + \frac{3}{x-2}\) for \(x > 2\).
The function g is defined by \(g(x) = 2x - 2\) for \(x > 0\).
Obtain a simplified expression for \(gf(x)\).
Solution
To find \(gf(x)\), substitute \(f(x)\) into \(g(x)\):
\(gf(x) = g(f(x)) = g\left(1 + \frac{3}{x-2}\right)\).
Substitute into \(g(x) = 2x - 2\):
\(gf(x) = 2\left(1 + \frac{3}{x-2}\right) - 2\).
Simplify the expression:
\(gf(x) = 2 + \frac{6}{x-2} - 2\).
Combine like terms:
\(gf(x) = \frac{6}{x-2}\).
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